Integrand size = 24, antiderivative size = 154 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {2255 \sqrt {1-2 x} (3+5 x)^2}{378 (2+3 x)}-\frac {(1-2 x)^{5/2} (3+5 x)^3}{12 (2+3 x)^4}+\frac {55 (1-2 x)^{3/2} (3+5 x)^3}{54 (2+3 x)^3}-\frac {55 \sqrt {1-2 x} (3+5 x)^3}{24 (2+3 x)^2}+\frac {275 \sqrt {1-2 x} (1123+4595 x)}{13608}+\frac {645865 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{6804 \sqrt {21}} \]
-1/12*(1-2*x)^(5/2)*(3+5*x)^3/(2+3*x)^4+55/54*(1-2*x)^(3/2)*(3+5*x)^3/(2+3 *x)^3+645865/142884*arctanh(1/7*21^(1/2)*(1-2*x)^(1/2))*21^(1/2)-2255/378* (3+5*x)^2*(1-2*x)^(1/2)/(2+3*x)-55/24*(3+5*x)^3*(1-2*x)^(1/2)/(2+3*x)^2+27 5/13608*(1123+4595*x)*(1-2*x)^(1/2)
Time = 0.23 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.49 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {\frac {21 \sqrt {1-2 x} \left (-3553918-19526798 x-39158517 x^2-32946525 x^3-8215200 x^4+1512000 x^5\right )}{2 (2+3 x)^4}+645865 \sqrt {21} \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{142884} \]
((21*Sqrt[1 - 2*x]*(-3553918 - 19526798*x - 39158517*x^2 - 32946525*x^3 - 8215200*x^4 + 1512000*x^5))/(2*(2 + 3*x)^4) + 645865*Sqrt[21]*ArcTanh[Sqrt [3/7]*Sqrt[1 - 2*x]])/142884
Time = 0.25 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {108, 27, 166, 27, 166, 27, 166, 164, 73, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^3}{(3 x+2)^5} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {1}{12} \int -\frac {55 (1-2 x)^{3/2} x (5 x+3)^2}{(3 x+2)^4}dx-\frac {(1-2 x)^{5/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {55}{12} \int \frac {(1-2 x)^{3/2} x (5 x+3)^2}{(3 x+2)^4}dx-\frac {(1-2 x)^{5/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {55}{12} \left (-\frac {1}{9} \int \frac {3 \sqrt {1-2 x} (5 x+3)^2 (12 x+5)}{(3 x+2)^3}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {55}{12} \left (-\frac {1}{3} \int \frac {\sqrt {1-2 x} (5 x+3)^2 (12 x+5)}{(3 x+2)^3}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {55}{12} \left (\frac {1}{3} \left (\frac {1}{6} \int -\frac {3 (14-61 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^2}dx+\frac {3 \sqrt {1-2 x} (5 x+3)^3}{2 (3 x+2)^2}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {55}{12} \left (\frac {1}{3} \left (\frac {3 \sqrt {1-2 x} (5 x+3)^3}{2 (3 x+2)^2}-\frac {1}{2} \int \frac {(14-61 x) (5 x+3)^2}{\sqrt {1-2 x} (3 x+2)^2}dx\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 166 |
| \(\displaystyle -\frac {55}{12} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {164 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)}-\frac {1}{21} \int \frac {(851-4595 x) (5 x+3)}{\sqrt {1-2 x} (3 x+2)}dx\right )+\frac {3 \sqrt {1-2 x} (5 x+3)^3}{2 (3 x+2)^2}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 164 |
| \(\displaystyle -\frac {55}{12} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {1}{21} \left (\frac {11743}{9} \int \frac {1}{\sqrt {1-2 x} (3 x+2)}dx-\frac {5}{9} \sqrt {1-2 x} (4595 x+1123)\right )+\frac {164 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)}\right )+\frac {3 \sqrt {1-2 x} (5 x+3)^3}{2 (3 x+2)^2}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {55}{12} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {1}{21} \left (-\frac {11743}{9} \int \frac {1}{\frac {7}{2}-\frac {3}{2} (1-2 x)}d\sqrt {1-2 x}-\frac {5}{9} \sqrt {1-2 x} (4595 x+1123)\right )+\frac {164 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)}\right )+\frac {3 \sqrt {1-2 x} (5 x+3)^3}{2 (3 x+2)^2}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {55}{12} \left (\frac {1}{3} \left (\frac {1}{2} \left (\frac {1}{21} \left (-\frac {23486 \text {arctanh}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )}{9 \sqrt {21}}-\frac {5}{9} \sqrt {1-2 x} (4595 x+1123)\right )+\frac {164 \sqrt {1-2 x} (5 x+3)^2}{21 (3 x+2)}\right )+\frac {3 \sqrt {1-2 x} (5 x+3)^3}{2 (3 x+2)^2}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^3}{9 (3 x+2)^3}\right )-\frac {(1-2 x)^{5/2} (5 x+3)^3}{12 (3 x+2)^4}\) |
-1/12*((1 - 2*x)^(5/2)*(3 + 5*x)^3)/(2 + 3*x)^4 - (55*((-2*(1 - 2*x)^(3/2) *(3 + 5*x)^3)/(9*(2 + 3*x)^3) + ((3*Sqrt[1 - 2*x]*(3 + 5*x)^3)/(2*(2 + 3*x )^2) + ((164*Sqrt[1 - 2*x]*(3 + 5*x)^2)/(21*(2 + 3*x)) + ((-5*Sqrt[1 - 2*x ]*(1123 + 4595*x))/9 - (23486*ArcTanh[Sqrt[3/7]*Sqrt[1 - 2*x]])/(9*Sqrt[21 ]))/21)/2)/3))/12
3.20.64.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_ ))*((g_.) + (h_.)*(x_)), x_] :> Simp[(-(a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x))*(a + b*x)^(m + 1)*(( c + d*x)^(n + 1)/(b^2*d^2*(m + n + 2)*(m + n + 3))), x] + Simp[(a^2*d^2*f*h *(n + 1)*(n + 2) + a*b*d*(n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)) Int[( a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && ILtQ[m, -1] && GtQ[n, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Time = 3.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.43
| method | result | size |
| risch | \(-\frac {3024000 x^{6}-17942400 x^{5}-57677850 x^{4}-45370509 x^{3}+104921 x^{2}+12418962 x +3553918}{13608 \left (2+3 x \right )^{4} \sqrt {1-2 x}}+\frac {645865 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{142884}\) | \(66\) |
| pseudoelliptic | \(\frac {1291730 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \left (2+3 x \right )^{4} \sqrt {21}+21 \sqrt {1-2 x}\, \left (1512000 x^{5}-8215200 x^{4}-32946525 x^{3}-39158517 x^{2}-19526798 x -3553918\right )}{285768 \left (2+3 x \right )^{4}}\) | \(70\) |
| derivativedivides | \(-\frac {500 \left (1-2 x \right )^{\frac {3}{2}}}{729}-\frac {7600 \sqrt {1-2 x}}{729}-\frac {4 \left (-\frac {159975 \left (1-2 x \right )^{\frac {7}{2}}}{112}+\frac {4220087 \left (1-2 x \right )^{\frac {5}{2}}}{432}-\frac {28870415 \left (1-2 x \right )^{\frac {3}{2}}}{1296}+\frac {21951755 \sqrt {1-2 x}}{1296}\right )}{9 \left (-4-6 x \right )^{4}}+\frac {645865 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{142884}\) | \(84\) |
| default | \(-\frac {500 \left (1-2 x \right )^{\frac {3}{2}}}{729}-\frac {7600 \sqrt {1-2 x}}{729}-\frac {4 \left (-\frac {159975 \left (1-2 x \right )^{\frac {7}{2}}}{112}+\frac {4220087 \left (1-2 x \right )^{\frac {5}{2}}}{432}-\frac {28870415 \left (1-2 x \right )^{\frac {3}{2}}}{1296}+\frac {21951755 \sqrt {1-2 x}}{1296}\right )}{9 \left (-4-6 x \right )^{4}}+\frac {645865 \,\operatorname {arctanh}\left (\frac {\sqrt {21}\, \sqrt {1-2 x}}{7}\right ) \sqrt {21}}{142884}\) | \(84\) |
| trager | \(\frac {\left (1512000 x^{5}-8215200 x^{4}-32946525 x^{3}-39158517 x^{2}-19526798 x -3553918\right ) \sqrt {1-2 x}}{13608 \left (2+3 x \right )^{4}}-\frac {645865 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right ) x -5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-21\right )+21 \sqrt {1-2 x}}{2+3 x}\right )}{285768}\) | \(87\) |
-1/13608*(3024000*x^6-17942400*x^5-57677850*x^4-45370509*x^3+104921*x^2+12 418962*x+3553918)/(2+3*x)^4/(1-2*x)^(1/2)+645865/142884*arctanh(1/7*21^(1/ 2)*(1-2*x)^(1/2))*21^(1/2)
Time = 0.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^5} \, dx=\frac {645865 \, \sqrt {21} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \log \left (\frac {3 \, x - \sqrt {21} \sqrt {-2 \, x + 1} - 5}{3 \, x + 2}\right ) + 21 \, {\left (1512000 \, x^{5} - 8215200 \, x^{4} - 32946525 \, x^{3} - 39158517 \, x^{2} - 19526798 \, x - 3553918\right )} \sqrt {-2 \, x + 1}}{285768 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]
1/285768*(645865*sqrt(21)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*log((3* x - sqrt(21)*sqrt(-2*x + 1) - 5)/(3*x + 2)) + 21*(1512000*x^5 - 8215200*x^ 4 - 32946525*x^3 - 39158517*x^2 - 19526798*x - 3553918)*sqrt(-2*x + 1))/(8 1*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^5} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.83 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {500}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {645865}{285768} \, \sqrt {21} \log \left (-\frac {\sqrt {21} - 3 \, \sqrt {-2 \, x + 1}}{\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}}\right ) - \frac {7600}{729} \, \sqrt {-2 \, x + 1} + \frac {12957975 \, {\left (-2 \, x + 1\right )}^{\frac {7}{2}} - 88621827 \, {\left (-2 \, x + 1\right )}^{\frac {5}{2}} + 202092905 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - 153662285 \, \sqrt {-2 \, x + 1}}{20412 \, {\left (81 \, {\left (2 \, x - 1\right )}^{4} + 756 \, {\left (2 \, x - 1\right )}^{3} + 2646 \, {\left (2 \, x - 1\right )}^{2} + 8232 \, x - 1715\right )}} \]
-500/729*(-2*x + 1)^(3/2) - 645865/285768*sqrt(21)*log(-(sqrt(21) - 3*sqrt (-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 7600/729*sqrt(-2*x + 1) + 1/2 0412*(12957975*(-2*x + 1)^(7/2) - 88621827*(-2*x + 1)^(5/2) + 202092905*(- 2*x + 1)^(3/2) - 153662285*sqrt(-2*x + 1))/(81*(2*x - 1)^4 + 756*(2*x - 1) ^3 + 2646*(2*x - 1)^2 + 8232*x - 1715)
Time = 0.28 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.77 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {500}{729} \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} - \frac {645865}{285768} \, \sqrt {21} \log \left (\frac {{\left | -2 \, \sqrt {21} + 6 \, \sqrt {-2 \, x + 1} \right |}}{2 \, {\left (\sqrt {21} + 3 \, \sqrt {-2 \, x + 1}\right )}}\right ) - \frac {7600}{729} \, \sqrt {-2 \, x + 1} - \frac {12957975 \, {\left (2 \, x - 1\right )}^{3} \sqrt {-2 \, x + 1} + 88621827 \, {\left (2 \, x - 1\right )}^{2} \sqrt {-2 \, x + 1} - 202092905 \, {\left (-2 \, x + 1\right )}^{\frac {3}{2}} + 153662285 \, \sqrt {-2 \, x + 1}}{326592 \, {\left (3 \, x + 2\right )}^{4}} \]
-500/729*(-2*x + 1)^(3/2) - 645865/285768*sqrt(21)*log(1/2*abs(-2*sqrt(21) + 6*sqrt(-2*x + 1))/(sqrt(21) + 3*sqrt(-2*x + 1))) - 7600/729*sqrt(-2*x + 1) - 1/326592*(12957975*(2*x - 1)^3*sqrt(-2*x + 1) + 88621827*(2*x - 1)^2 *sqrt(-2*x + 1) - 202092905*(-2*x + 1)^(3/2) + 153662285*sqrt(-2*x + 1))/( 3*x + 2)^4
Time = 0.07 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.71 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^3}{(2+3 x)^5} \, dx=-\frac {7600\,\sqrt {1-2\,x}}{729}-\frac {500\,{\left (1-2\,x\right )}^{3/2}}{729}-\frac {\frac {21951755\,\sqrt {1-2\,x}}{236196}-\frac {28870415\,{\left (1-2\,x\right )}^{3/2}}{236196}+\frac {4220087\,{\left (1-2\,x\right )}^{5/2}}{78732}-\frac {1975\,{\left (1-2\,x\right )}^{7/2}}{252}}{\frac {2744\,x}{27}+\frac {98\,{\left (2\,x-1\right )}^2}{3}+\frac {28\,{\left (2\,x-1\right )}^3}{3}+{\left (2\,x-1\right )}^4-\frac {1715}{81}}-\frac {\sqrt {21}\,\mathrm {atan}\left (\frac {\sqrt {21}\,\sqrt {1-2\,x}\,1{}\mathrm {i}}{7}\right )\,645865{}\mathrm {i}}{142884} \]
- (21^(1/2)*atan((21^(1/2)*(1 - 2*x)^(1/2)*1i)/7)*645865i)/142884 - (7600* (1 - 2*x)^(1/2))/729 - (500*(1 - 2*x)^(3/2))/729 - ((21951755*(1 - 2*x)^(1 /2))/236196 - (28870415*(1 - 2*x)^(3/2))/236196 + (4220087*(1 - 2*x)^(5/2) )/78732 - (1975*(1 - 2*x)^(7/2))/252)/((2744*x)/27 + (98*(2*x - 1)^2)/3 + (28*(2*x - 1)^3)/3 + (2*x - 1)^4 - 1715/81)